Certainly! Let's solve each expression step-by-step using the rules of exponents.
### 1. Simplify [tex]\((2x^4)(6x^4)\)[/tex]
To simplify the expression [tex]\((2x^4)(6x^4)\)[/tex], we use the property of exponents which states that when we multiply like bases, we add the exponents. Additionally, we multiply the coefficients directly.
[tex]\[
(2x^4)(6x^4) = 2 \times 6 \times x^{4+4} = 12x^8
\][/tex]
### 2. Simplify [tex]\((7x^4)^2\)[/tex]
To simplify the expression [tex]\((7x^4)^2\)[/tex], we use the power of a power property, which states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Here, we need to raise both the coefficient and the variable to the power of 2.
[tex]\[
(7x^4)^2 = 7^2 \times (x^4)^2 = 49 \times x^{4 \cdot 2} = 49x^8
\][/tex]
### 3. Simplify [tex]\(\frac{15x^6}{5x^4}\)[/tex]
To simplify the fraction [tex]\(\frac{15x^6}{5x^4}\)[/tex], we use the property of exponents which states that when we divide like bases, we subtract the exponents.
[tex]\[
\frac{15x^6}{5x^4} = \frac{15}{5} \times x^{6-4} = 3 \times x^2 = 3x^2
\][/tex]
### 4. Simplify [tex]\(7x^{-3}\)[/tex]
To simplify [tex]\(7x^{-3}\)[/tex], we use the property of exponents which states that a negative exponent indicates a reciprocal. Specifically, [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex].
[tex]\[
7x^{-3} = 7 \times \frac{1}{x^3} = \frac{7}{x^3}
\][/tex]
### Summary
- [tex]\(\left(2x^4\right)\left(6x^4\right) = 12x^8\)[/tex]
- [tex]\(\left(7x^4\right)^2 = 49x^8\)[/tex]
- [tex]\(\frac{15x^6}{5x^4} = 3x^2\)[/tex]
- [tex]\(7x^{-3} = \frac{7}{x^3}\)[/tex]
